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arXiv · 2609.03455

The Klein-Vélu septic, 2-Division, and modular function fields of level 14

Masanobu Kaneko·Masato Kuwata
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Abstract

We study the 2-division of the Klein-Vélu elliptic normal septic. Its three non-zero 2-torsion points give rise to an explicit cubic equation over the function field of $X(7)$. We identify the three roots of this cubic with modular functions expressed in terms of septic theta functions at $τ/2,τ$, and $2τ$, and show that its splitting field is precisely the function field of $X(14)$. This gives an explicit link between the 2-division geometry of the Klein-Vélu septic and the passage from full level 7 to full level $14$. Several intermediate modular function fields in the resulting $S_3$-extension are also described.

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Masanobu Kaneko, Masato Kuwata. 2026-09-03. The Klein-Vélu septic, 2-Division, and modular function fields of level 14. https://arxiv.org/abs/2609.03455

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